The Central Limit Theorem The Central Limit Theorem andand

the Normal Distributionthe Normal Distribution

Recapitulation from Last TimeRecapitulation from Last Time

1. Statistical inference involves generalizing from a sample to a (statistical) universe.2. Statistical inference is only possible with random samples.3. Statistical inference estimates the probability that a sample result could be due to chance (in the selection of the sample).4. Sampling distributions are the keys that connect (known) sample statistics and (unknown) universe parameters.5. Alpha (significance) levels are used to identify critical values on sampling distributions.

The Central Limit TheoremThe Central Limit TheoremIf repeated random samples of size N are drawn from a population that is normally distributed along some variable Y, having a mean and a standard deviation , then the sampling distribution of all theoretically possible sample means will be a normal distribution having a mean and a standard deviation given by

[Sirkin (1999), p. 239]

n

sY

22 2/)(

22

1YYY

Y

Yp

Mathematically,

A normal distribution:

1. is symmetrical (both halves are identical);2. is asymptotic (its tails never touch the underlying x-axis; the curve reaches to – and + and thus must be truncated); 3. has fixed and known areas under the curve (these fixed areas are marked off by units along the x-axis called z-scores; imposing truncation, the normal curve ends at + 3.00 z on the right and - 3.00 z on the left).

A normal distribution:

1. is symmetrical (both halves are identical);2. is asymptotic (its tails never touch the underlying x-axis; the curve reaches to – and + and thus must be truncated); 3. has fixed and known areas under the curve (these fixed areas are marked off by units along the x-axis called z-scores; imposing truncation, the normal curve ends at + 3.00 z on the right and - 3.00 z on the left).

MeanMean Standard DeviationStandard Deviation Variance Variance

UniverseUniverse Y Y Y2

SamplingSampling Y

DistributionDistribution _

SampleSample Y sY sY2

  

2ˆYY

N

sY

The Standard ErrorThe Standard Error

where sY = sample standard deviation and N = sample size

Let's assume that we have a random sample of 200 USC undergraduates. Note that this is both a large and a random sample, hence the Central Limit Theorem applies to any statistic that we calculate from it. Let's pretend that we asked these 200 randomly-selected USC students to tell us their grade point average (GPA). (Note that our statistical calculations assume that all 200 [a] knew their current GPA and [b] were telling the truth about it.) We calculated the mean GPA for the sample and found it to be 2.58. Next, we calculated the standard deviation for these self-reported GPA values and found it to be 0.44.

The standard error is nothing more than the standard deviation of the sampling distribution. The Central Limit Theorem tells us how to estimate it:

N

sY

The standard error is estimated by dividing the standard deviation of the sample by the square root of the size of the sample. In our example,

200

44.0ˆ

142.14

44.0ˆ

031.0ˆ

RecapitulationRecapitulation

1. The Central Limit Theorem holds only for large, random samples.2. When the Central Limit Theorem holds, the mean of the sampling distribution is equal to the mean in the universe (also ). 3. When the Central Limit Theorem holds, the standard deviation of the sampling distribution (called the standard error, ) is estimated by

N

sY

Y

Recapitulation (continued)Recapitulation (continued)

4. When the Central Limit Theorem holds, the sampling distribution is normally shaped. 5. All normal distributions are symmetrical, asymptotic, and have areas that are fixed and known.